In the 2020–21 season, with the ongoing COVID-19 pandemic affecting the commencement of the collegiate season, Drury opted to leave Harvard and signed a one-year deal with Swedish club, the Växjö Lakers of the Swedish Hockey League (SHL).

He continued his impressive play in the post-season, finishing second in the league with 11 points in 14 playoff games as Växjö Lakers won the Le Mat Trophy.

[3] On July 8, 2021, Drury returned to North America in agreeing to a three-year, entry-level contract with his draft club, the Carolina Hurricanes.

[5] On January 24, 2025, Drury, Martin Nečas, a 2025 second-round pick, and a 2026 fourth-round pick were traded by the Hurricanes to the Colorado Avalanche in a three-team deal involving the Chicago Blackhawks; the other sides of the deal included the Hurricanes receiving Mikko Rantanen and Taylor Hall, while the Blackhawks received Carolina's 2025 third-round pick as compensation for retaining 50% of Rantanen's remaining contract.

[1] His uncle is former professional hockey player Chris Drury, who currently serves as general manager for the New York Rangers.